QUANTUM MECHANICS. Yehuda B. Band. and Yshai Avishai WITH APPLICATIONS TO NANOTECHNOLOGY AND INFORMATION SCIENCE

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1 QUANTUM MECHANICS WITH APPLICATIONS TO NANOTECHNOLOGY AND INFORMATION SCIENCE Yehuda B. Band Department of Chemistry, Department of Electro-Optics and Department of Physics, and Use Katz Institute for Nanoscale Science and Technology Ben-Gurion University Beer-Sheva, Israel and Yshai Avishai Department of Physics, and Use Katz Institute for Nanoscale Science and Technology Ben-Gurion University Beer-Sheva, Israel m* '~Wm f^f P j Jfcvl&Hte ELSEVIER AMSTERDAM WALTHAM HEIDELBERG LONDON NEW YORK OXFORD PARIS SAN DIEGO SAN FRANCISCO SINGAPORE SYDNEY TOKYO Academic Press is an imprint of Elsevier

2 CONTENTS Preface xvii CHAPTER 1 INTRODUCTION TO QUANTUM MECHANICS What is Quantum Mechanics? A Brief Early History of Quantum Mechanics Energy Quantization Waves, Light, and Blackbody Radiation Wave-Particle Duality Angular Momentum Quantization Tunneling Photoelectric Effect Nanotechnology and Information Technology STM and AFM Microscopies Molecular Electronics Quantum Dots, Wires and Wells, and Nanotubes Bio-Nanotechnology Information Technology A First Taste of Quantum Mechanics Quantum States and Probability Distributions Observable Operators Quantum Entanglement The Postulates of Quantum Mechanics Time-Dependent and -Independent Schrodinger Equations Momentum, Energy, and Angular Momentum Dirac Delta Functions Position and Momentum States, x> and p> Ehrenfest's Theorem One-Dimensional Wave Equations Particle-in-a-Box and Piecewise-Constant Potentials The Delta Function Potential Wave Packets The Linear Potential and Quantum Tunneling The Harmonic Oscillator 55

3 vi CHAPTER 2 THE FORMALISM OF QUANTUM MECHANICS Hubert Space and Dirac Notation Position and Momentum Representations Basis-State Expansions Hermitian and Anti-Hermitian Operators Compatible Operators and Degeneracy The Uncertainty Principle The Measurement Problem Mixed States: Density Matrix Formulation Many-Particle Systems: Correlation Functions Purity and von Neumann Entropy Distance Between States The Measurement Problem Revisited The Wigner Representation Schrödinger and Heisenberg Representations Interaction Representation Harmonic Oscillator Raising-Lowering Operators Coherent States and Squeezed States The Correspondence Principle and the Classical Limit Symmetry and Conservation Laws in Quantum Mechanics Exchange Symmetry Inversion Symmetry Time-Reversal Symmetry Additional Generators of Galilean Transformations 102 CHAPTER 3 ANGULAR MOMENTUM AND SPHERICAL SYMMETRY Angular Momentum in Quantum Mechanics Angular Momentum Raising and Lowering Operators Electron Spin: j= 1/ Angular Momentum in Spherical Coordinates Spherical Harmonics Spherically Symmetric Systems Angular Momentum Decomposition of Plane Waves Spherical Quantum Dot The 3D Harmonic Oscillator The Morse Oscillator van der Waals and Lennard-Iones Potentials The Hydrogen Atom Rotations and Angular Momentum Euler angles a, ß, y and the Rotation Matrix Rotation and D Functions Rigid-Rotor Eigenfunctions Addition (Coupling) of Angular Momenta Clebsch-Gordan Coefficients and 3j Symbols Clebsch-Gordan Series 143

4 Contents vii 3.5 Tensor Operators Irreducible Representations of the Density Matrix Vector Fields Spinor Fields Multipole Expansions Symmetry Considerations Selection Rules Inversion Symmetry Time-Reversal Symmetry Wigner-Eckart Theorem / and Higher Coefficients 156 CHAPTER 4 SPIN Spin Angular Momentum Spinors Pauli Matrices Rotation of Spinors Spin-Orbitals Electron in a Magnetic Field Charged Particle in a Magnetic Field: Orbital Effects Time-Reversal Properties of Spinors Spin-Orbit Interaction in Atoms Hyperfine Interaction Electric Quadrupole Hyperfine Interaction Zeeman Splitting of Hyperfine States Spin-Dipolar Interactions Introduction to Magnetic Resonance The Rotating-Wave Approximation Spin Relaxation and the Bloch Equation Nuclear Spin Hamiltonian Chemical Shifts Fourier Transform NMR 190 CHAPTER 5 QUANTUM INFORMATION Classical Computation and Classical Information Information and Entropy Shannon Entropy Data Compression Classical Computers and Gates Classical Cryptography Computational Complexity Quantum Information Qubits Quantum Entanglement and Bell States Quantum Gates 213

5 5.2.4 No-Cloning Theorem Dense Coding Data Compression of Quantum Information Quantum Teleportation Quantum Cryptography Quantum Circuits Quantum Computing Despite Measurement Quantum Computing Algorithms Deutsch and Deutsch-Jozsa Algorithms The Grover Search Algorithm Quantum Fourier Transform Shor Factorization Algorithm Quantum Simulation Decoherence Quantum Error Correction Experimental Implementations Ion Traps Neutral Atoms in Optical Lattices Cavity Based Quantum Computing Nuclear Magnetic Resonance Systems All-Optical Quantum Computers Solid-State Qubits The EPR Paradox Bell's Inequalities Bell's Inequalities and the EPR Paradox Bell's Analysis using Hidden Variables General Aspects of Bell's Inequalities 256 CHAPTER 6 QUANTUM DYNAMICS AND CORRELATIONS Two-Level Systems Two-Level Dynamics (Spin Dynamics) The Bloch Sphere Picture Coupling to a Bath: Decoherence Periodically Driven Two-Level System Atoms in an Electromagnetic Field: Dispersion and Absorption Doppler Cooling of Atoms Optical Trapping of Atoms Two or More Correlated "Spins" The TV-Two-Level System Bloch Sphere Ramsey Fringe Spectroscopy Three-Level Systems Two or More Three-Level Correlated Systems Classification of Correlation and Entanglement Entanglement Witness Operators Three-Level System Dynamics Continuous-Variable Systems 295

6 Contents 6.6 Wave Packet Dynamics Time-Dependent Hamiltonians Quantum Optimal Control Theory 300 CHAPTER 7 APPROXIMATION METHODS Basis-State Expansions Time-Dependent Basis Set Expansions Semiclassical Approximations The WKB Approximation Semiclassical Treatment of Dynamics Semiclassical Hamilton-Jacobi Expansion Perturbation Theory Nondegenerate Perturbation Theory Degenerate Perturbation Theory Time-Dependent Perturbation Theory Dynamics in an Electromagnetic Field Spontaneous and Stimulated Emission of Radiation Electric Dipole and Multipole Radiation Thomson, Rayleigh, Raman, and Brillouin Transitions Decay Width Doppler Shift Exponential and Nonexponential Decay The Variational Method The Sudden Approximation The Adiabatic Approximation Chirped Pulse Adiabatic Passage Stimulated Raman Adiabatic Passage The Landau-Zener Problem Generalized Displacements and Forces Berry Phase Linear Response Theory Susceptibilities Kubo Formulas Onsager Reciprocal Relations Fluctuation-Dissipation Theorem 362 CHAPTER 8 IDENTICAL PARTICLES Permutation Symmetry The Symmetric Group % Young Tableaux Exchange Symmetry Symmetrization Postulate Permanents and Slater Determinants Simple Two- and Three-Electron States Exchange Symmetry for Two Two-Level Systems Many-Particle Exchange Symmetry 378

7 X Contents CHAPTER 9 ELECTRONIC PROPERTIES OF SOLIDS The Free Electron Gas Density of States in 2D and ID Systems Fermi-Dirac Distribution Elementary Theories of Conductivity Drude Theory of Conductivity Thermal Conductivity of Metals Sommerfeld Theory of Transport in Metals Crystal Structure Bravais Lattices and Crystal Systems The Reciprocal Lattice Quasicrystals Electrons in a Periodic Potential From Atomic Orbits to Band Structure Band Structure and Electron Transport Periodic Potential and Band Formation Bloch Wave Functions and Energy Bands Schrödinger Equation in Reciprocal Lattice Space Sinusoidal Potential: Mathieu Functions Tight-Binding Model Wannier Functions Electric Field Effects Magnetic Field Effects Electron in a Magnetic Field Aharonov-Bohm Effect The Hall Effect and Magnetoresistance Landau Quantization D Electron Gas in a Perpendicular Magnetic Field Electron Subject to Periodic Potential and Magnetic Field de Haas-van Alphen and Shubnikov-de Haas Effects The Quantum Hall Effect Paramagnetism and Diamagnetism Magnetic Order Semiconductors Semiconductor Band Structure Charge Carrier Density Extrinsic Semiconductors Inhomogeneous Semiconductors: p-n Junctions Excitons Spin-Orbit Coupling in Solids k p Perturbation Theory Spin Hall Effect Photon Induced Processes in Semiconductors Spintronics Tools for Manipulating Spins Tunneling Magnetoresistance Spintronic Devices 523

8 Contents 9.8 Low-Energy Excitations Phonons Plasmons Magnons Polarons Polaritons Insulators Denning Insulators Classification of Insulators 543 CHAPTER 10 ELECTRONIC STRUCTURE OF MULTIELECTRON SYSTEMS 10.1 The Multielectron System Hamiltonian Slater and Gaussian Type Atomic Orbitals Term Symbols for Atoms Two-Electron Systems The Helium Atom The Hartree Method: Helium Hartree Approximation for Multielectron Systems The Hartree-Fock Method Hartree-Fock for Helium Koopmans' Theorem Atomic Radii Multielectron Fine Structure: Hund's Rules Electronic Structure of Molecules //^Molecular Orbitals The Hydrogen Molecule The Hiickel Approximation Hartree-Fock for Metals Electron Correlation Configuration Interaction Moller-Plesset Many-Body Perturbation Theory Coupled Cluster Method 577 CHAPTER 11 MOLECULES Molecular Symmetries Molecular Orbitals and Group Theory Character Tables and Mulliken Symbols Diatomic Electronic States Interatomic Potentials at Large Internuclear Distances Hund's Coupling Hyperfine Interactions in Diatomic Molecules The Born-Oppenheimer Approximation Potential Energy Crossings and Pseudocrossings Born-Oppenheimer Nuclear Derivative Coupling The Hellman-Feynman Theorem Rotational and Vibrational Structure 593

9 XII 11.5 Vibrational Modes and Symmetry Selection Rules for Optical Transitions Selection Rules for Diatomic Molecules The Franck-Condon Principle Bound-Free Matrix Elements 602 CHAPTER 12 SCATTERING THEORY Classical Scattering Theory Quantum Scattering Time-Dependent and Stationary Approaches Preparation of the Initial State Time-Dependent Formulation Stationary Scattering Theory Cross-Sections Two-Body Collisions From Wave Functions to Cross-Sections Green's Function Aspects of Formal Scattering Theory The Transition Operator (The T Matrix) The S Matrix and Möller Operators Central Potentials Central Potentials and Spin Axial Symmetry Partial Wave Analysis Phase Shift Analysis Scattering from a Coulomb Potential Scattering of Two Identical Particles Resonance Scattering Influence of Bound States Resonance Cross-Sections Feshbach Resonance Fano Resonance Approximation Methods Born Approximation WKB Approximation The Variational Principle Eikonal Approximation Particles with Internal Degrees of Freedom Spin Composite Particles Channels Asymptotic States and Cross-Sections Möller Operators The Multichannel S Matrix Scattering from Two Potentials Scattering of Particles with Spin 704

10 Contents xiii Inelastic Scattering and Scattering Reactions Scattering from a Collection of Identical Particles Scattering in Low-Dimensional Systems Scattering in Two Dimensions ID Scattering: S Matrix ID Scattering: Anderson Localization Scattering in Quasi-One-Dimensional Systems 741 CHAPTER 13 LOW-DIMENSIONAL QUANTUM SYSTEMS Mesoscopic Systems Low-Dimensional Nanostructures Quantum Wells Quantum Wires Quantum Dots Heterojunctions and Superlattices Quantum Point Contacts The Landauer Conductance Formula Aharonov-Bohm Interferometer Multiport Landauer Formula Conductance of Quantum Point Contacts Properties of Quantum Dots Equilibrium Properties of Quantum Dots Charge Transport through Quantum Dots Disorder in Mesoscopic Systems Disorder in Quantum Dots Disordered Systems and Random Matrices Application of Random Matrix Theory to Disordered Mesoscopic Systems Kondo Effect in Quantum Dots The Dot Kondo Hamiltonian The Dot Kondo Temperature Abrikosov-Suhl Resonance and Kondo Conductance Graphene Carbon Nanotubes Lattice Structure and Dirac Cones Dirac Equation and its Relevance to Graphene Tight-Binding Model for Graphene Continuum Theory Landau Levels in Graphene Potential Scattering in Graphene Inventory of Recently Discovered Low-Dimensional Phenomena Persistent Currents Weak Localization Shot Noise Strongly Correlated Low-Dimensional Systems Wigner Crystals The Fractional Quantum Hall Effect 816

11 XIV High-Temperature Superconductivity and 2D Magnetism ID Correlated Electron Systems ID Spin Chains Quantum Spin Hall Effect 822 CHAPTER 14 MANY-BODY THEORY Second Quantization Interacting Identical Particles: A Generic Many-Body Problem A Basis for Many-Body Wave Functions Mapping onto Fock Space Creation and Annihilation Operators The Hamiltonian in Fock Space Field Operators Electromagnetic Field Quantization: Photons Quantization of Lattice Vibrations: Phonons Systems with Two Kinds of Particles Statistical Mechanics in Second Quantization The Electron Gas Electrons in the Jellium Model Mean-Field Theory Mean-Field Equations in Second Quantization 867 CHAPTER 15 DENSITY FUNCTIONAL THEORY The Hohenberg-Kohn Theorems The Thomas-Fermi Approximation The Kohn-Sham Equations Local Density Approximation Spin DFT and Magnetic Systems Spin DFT Local Density Approximation The Gap Problem in DFT Time-Dependent DFT The Runge-Gross Theorem Time-Dependent Kohn-Sham Equations AdiabaticLDA DFT Computer Packages 888 APPENDIX A LINEAR ALGEBRA 891 A.l Vector Spaces 891 A. 1.1 Dirac Notation 892 A. 1.2 Inner Product Spaces 893 A.2 Operators and Matrices 898 A.2.1 Outer Product 899 A.2.2 Determinants and Permanents 900 A.2.3 Normal, Unitary, Hermitian-Conjugate and Hermitian Operators A.2.4 Trace and Projection 902 A.2.5 Antilinear and Antiunitary Operators 903

12 Contents APPENDIX B SOME ORDINARY DIFFERENTIAL EQUATIONS APPENDIX C VECTOR ANALYSIS 913 C.l Scalar and Vector Products 913 C.2 Differential Operators 914 C.3 Divergence and Stokes Theorems 915 C.4 Curvilinear Coordinates 916 \PPENDIX D FOURIER ANALYSIS 921 D.l Fourier Series 922 D. 1.1 Fourier Series of Functions of a Discrete Variable 924 D.2 Fourier Integrals 925 D.3 Fourier Series and Integrals in Three-Space Dimensions 927 D.3.1 3D Fourier Integrals 927 D.4 Fourier Integrals of Time-Dependent Functions 928 D.5 Convolution 928 D.6 Fourier Expansion of Operators 929 D.7 Fourier Transforms 929 D.8 FT for Solving Differential and Integral Equations 931 \PPENDIX E SYMMETRY AND GROUP THEORY 933 E.l Group Theory Axioms 933 E.2 Group Multiplication Tables 933 E.3 Examples of Groups 934 E.3.1 Point Groups 935 E.3.2 Space Groups 936 E.3.3 Continuous Groups 936 E.4 Some Properties of Groups 936 E.5 Group Representations 937 E.5.1 Irreducible Representations 938 E.5.2 Group Orthogonality Theorem 939 E.5.3 Characters and Character Tables 939 E.5.4 Constructing Irreducible Representations 942 Bibliography 943 Index 953